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Computer Physics Communications 183 (2012) 2413–2423
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A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation Ram Jiwari ∗ School of Mathematics & Computer Applications, Thapar University, Patiala 147004, India
article
abstract
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Article history: Received 19 April 2012 Received in revised form 31 May 2012 Accepted 16 June 2012 Available online 19 June 2012
In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs. © 2012 Elsevier B.V. All rights reserved.
Keywords: Haar wavelets Burgers’ equation Multi-resolution analysis Collocation points
1. Introduction Consider the one-dimensional Burgers’ equation
∂u ∂ 2u ∂u −ν 2 +u = 0, ∂t ∂x ∂x
(x, t ) ∈ Ω × (0, T ],
(1)
with the initial condition u(x, 0) = f (x),
0 ≤ x ≤ 1,
(2a)
and the boundary conditions u(0, t ) = f1 (t ),
u(1, t ) = f2 (t ) 0 ≤ t ≤ T
(2b)
where Ω = (0, 1), ν > 0 is the coefficient of kinematic viscosity and the prescribed function f (x) is sufficiently smooth. The nonlinear homogeneous quasilinear parabolic partial differential equation is the simplest nonlinear model equation for diffusive waves in fluid dynamics. Burgers’ equation arises in many physical problems including one-dimensional turbulence, sound waves in a viscous medium, shock waves in a viscous medium, waves in fluid filled viscous elastic tubes, and magnetohydrodynamic waves in a medium with finite electrical conductivity. Such type of equation was first introduced by Bateman [1] in 1915 and he proposed the steady-state solution of the problem. In 1948, Burgers [2,3] introduced this equation to capture some features of turbulent fluid in a channel caused by the interaction of the
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opposite effects of convection and diffusion. Therefore, Eq. (1) is referred to as ‘‘Burgers’ equation’’. The structure of Burgers’ equation is roughly similar to that of Navier–Stoke’s equations due to the presence of the nonlinear convection term and the occurrence of the diffusion term with viscosity coefficient. So this equation can be considered as a simplified form of Navier–Stoke’s equations. The study of the general properties of Burgers’ equation has attracted attention of scientific community due to its applications in various fields such as gas dynamics, heat conduction, elasticity, etc. The study of the solution of Burgers’ equation has been carried out for last half Century and still it is an active area of research to develop some better numerical scheme to approximate its solution. So far various numerical algorithms such as the automatic differentiation method [4], Galerkin finite element method [5], cubic B-splines collocation method [6], spectral collocation method [7,8], Sinc Differential Quadrature Method [9], Polynomial based differential quadrature method [10], Quartic B-splines Differential Quadrature Method [11], Quartic B-splines collocation method [12], Quadratic B-splines finite element method [13], finite element method [14], Fourth-order finite difference method [15], a parameter-uniform implicit difference scheme [16], A novel numerical scheme [17], explicit and exactexplicit finite difference methods [18], least-squares quadratic B-splines finite element method [19], implicit fourth-order compact finite difference scheme [20], some implicit methods [21], Adomian–Pade technique [22], variational iteration method [23], homotopy analysis method [24], differential transform method and the homotopy analysis method [25], Semi-Implicit Finite Difference Schemes [26], modified cubic B-splines collocation method [27] etc.
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Wavelets have been used for the solution of partial differential equations (PDEs) since the 1980s. The good features of this approach are possibility to detect singularities, irregular structure and transient phenomena exhibited by the analyzed equations. Most of the wavelets algorithms for solving PDE are based on the collocation [28,29] or the Galerkin techniques method [28,30,31]. Evidently, all attempts to simplify the wavelet solutions for PDE are welcome; one of the possibilities for this is to make use of the Haar wavelet family. Haar wavelets (which are Daubechies of order 1) consist of piecewise constant functions and are therefore the simplest orthonormal wavelets with a compact support. The Haar wavelets have gained popularity among researchers for their useful properties such as simple applicability, orthogonality and compact support. Compact support of the Haar wavelet basis permits straight inclusion of the different types of boundary conditions in the numeric algorithms. A drawback of the Haar wavelets is their discontinuity. Since the derivatives do not exist in the breaking points it is not possible to apply the Haar wavelets for solving PDE directly. Lepik [32–34] had solved higher order as well as nonlinear ODEs by using the Haar wavelet method. Recently, Hariharan and his associates [35,36] have used Haar wavelets methods for the solution of some nonlinear PDEs. In this paper, an efficient numerical scheme based on uniform Haar wavelets and quasilinearization process is proposed for the numerical simulation of the nonlinear homogeneous quasilinear parabolic Burger’s equation. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents solutions of boundary value problems. The accuracy of the proposed scheme is demonstrated by three test problems. The numerical results are compared with existing numerical and exact solutions and it is found that the proposed scheme produce better results. The use of uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.
Wavelet transform or wavelet analysis is a recently developed mathematical tool for many problems. One of the popular families of wavelet is Haar wavelets. The Haar function is in fact the Daubechies wavelet of order 1. Due to its simplicity, the Haar wavelet had become an effective tool for solving many problems arising in many branches of sciences. Haar functions have been used since 1910. It was introduced by the Hungarian mathematician Alfred Haar. The Haar function is an odd rectangular pulse pair, is the simplest and oldest orthonormal wavelet with compact support. There are different definitions for the Haar function and various generalizations have been used and published. Haar showed that certain square wave function could be translated and scaled to create a basis set that span L2 . Year later, it was seen that the system of Haar is a particular wavelet system. If we choose scaling function to have compact support over 0 ≤ x ≤ 1, that is the Haar wavelet family for x ∈ [0, 1) is defined as hi ( x ) =
1
−1 0
h1 (x) =
x ∈ [0, 1), otherwise.
1 0
(5)
If we want to solve partial differential equations of any order, we need the following integrals pi,1 (x) =
x
hi (x′ )dx′ , 0
(6)
x
pi,v+1 (x) =
pi,v (x′ )dx′ ,
v = 1, 2, 3, . . . .
0
Taking into account (3) the integrals (6) can be calculated analytically; by doing so we obtained the following x − ξ1 ξ3 − x 0
pi,1 (x) =
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ), elsewhere
(x − ξ1 )2 2 1 (ξ3 − x)2 pi,2 (x) = 4m2 − 2 1 4m2 0
(7)
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ),
(8)
x ∈ [ξ1 , 1) elsewhere.
3. Function approximation Any function y(x) which is square integrable in the interval (0, 1) can be expressed in the following form of Haar wavelets y(x) =
∞
ai hi (x).
(9)
i=1
2. Haar wavelets
index i in Eq. (4) is calculated from the formula i = m + k + 1. In the case of minimal values m = 1, k = 0, we have i = 2. The maximum value of i is i = 2M = 2J +1 . For i = 1, the function h1 (x) is the scaling function for the family of the Haar wavelets which is defined as
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ), otherwise
(3)
The above series terminates at finite terms if y(x) is piecewise constant or can be approximated as piecewise constant during each subinterval, then y(x) will be terminated at finite terms, that is y(x) =
2M
ai hi (x) = aT(2M ) h(2M ) (x),
(10)
i=1
where the coefficients aT(2M ) and the Haar function vector h(2M ) (x) are defined as aT(2M ) = [a1 , a2 , . . . , a2M ]
and
h(2M ) (x) = [h(1) (x), h(2) (x), . . . , h(2M ) (x)]T where T denotes the transpose and M = 2j . The best way to understand wavelets is through a multiresolution analysis. Given a function y(x) ∈ L2 (R) a multi-resolution analysis (MRA) of L2 (R) produces a sequence of subspaces Uj , Uj+1 , . . . such that the projections of y(x) onto these spaces give finer and finer approximations of the function y(x) as j → ∞. The details of MRA is as follows. 3.1. Multi-resolution analysis
where (4)
A multi-resolution analysis of L2 (R) is defined as a sequence of closed subspaces Uj ⊂ L2 (R), j ∈ Z with the following properties.
In the above definition the integer m = 2j , j = 0, 1, . . . , J indicates the level of the wavelet and the integer k = 0, 1, . . . , m − 1 is the translation parameter. The maximal level of resolution is J. The
(i) · · · ⊂ U−1 ⊂ U0 ⊂ U1 ⊂ · · ·. 2 (ii) The space Uj satisfying j∈Z Uj is dense in L (R) and j∈Z Uj = 0.
ξ1 =
k m
,
ξ2 =
k + 0.5 m
,
ξ3 =
k+1 m
.
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
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Fig. 1. Physical behavior of numerical solutions of Example 1 for ν = 0.1 at different times t with 1t = 0.001.
Fig. 2. Physical behavior of numerical solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.001.
Fig. 3. Physical behavior of numerical solutions of Example 1 in 3D and contour for ν = 0.1 at different times t with 1t = 0.001.
(iii) If g (x) ∈ U0 , g (2j x) ∈ Uj i.e. the space Uj is a scaled version of the central space U0 . (iv) If g (x) ∈ U0 , g (2j x − k) ∈ Uj i.e. all the Uj are invariant under translation. (v) There exists Φ ∈ U0 such that Φ (x − k); k ∈ Z is a Riesz basis in U0 . The space Uj is used to approximate general functions by defining appropriate projection of these functions onto these spaces. Since
the union of all the Uj is dense in L2 (R), it guarantees that any function in L2 (R) can be approximated arbitrarily close by such projections. As an example the space Uj can be defined like J +1
Uj = Wj−1 ⊕ Uj−1 = Wj−1 ⊕ Wj−2 ⊕ Vj−2 = · · · = ⊕ Wj ⊕ V0 j =1
then the scaling function h1 (x) generates an MRA for the sequence of spaces { Uj , j ∈ Z } by translation and dilation as defined in Eqs. (3) and (5). For each j the space Wj serves as the orthogonal
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Fig. 4. Physical behavior of numerical solutions of Example 1 in 3D and contour for ν = 0.01 at different times t with 1t = 0.001.
Fig. 5. Physical behavior of numerical (left) and exact (right) solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.001. Table 1 Comparison with exact and existing numerical methods of Example 1 for ν = 0.1 at different times t and x. x
t
[18] 1t = 0.001
[14]
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.30891 0.24075 0.19568 0.16257 0.02720
0.31429 0.24373 0.19758 0.16391 0.02743
0.30881 0.24069 ... 0.16254 0.02720
0.30887 0.24070 0.19566 0.16255 0.02721
0.30889 0.24074 0.19568 0.16256 0.02720
0.50
0.4 0.6 0.8 1.0 3.0
0.56964 0.44721 0.35924 0.29192 0.04021
0.57636 0.45169 0.36245 0.29437 0.04057
0.56955 0.44714 ... 0.29188 0.04021
0.56956 0.44715 0.35920 0.29188 0.04022
0.56963 0.44721 0.35924 0.29192 0.04021
0.75
0.4 0.6 0.8 1.0 3.0
0.62542 0.48721 0.37392 0.28748 0.02977
0.62592 0.49034 0.37713 0.29016 0.01334
0.62540 0.48715 ... 0.28744 0.02978
0.62540 0.48716 0.37389 0.28743 0.02978
0.62544 0.48721 0.37392 0.28747 0.02977
complement of Uj in Uj+1 . The space Wj includes all the functions in Uj+1 that are orthogonal to all those in Uj under some chosen inner product. The set of functions which form basis for the space Wj are called wavelets [37,38].
Let us divide the interval (0, 1] into N equal parts of length 1t = 1/N and denote ts = (s − 1)1t , s = 1, 2, . . . , N. We assume the following for the solution of Eq. (1), assumed by Lepik [33] 2M
4. Uniform Haar wavelet based scheme for Burgers’ equation
u˙ ′′ (x, t ) =
as (i)hi (x) = aT(2M ) h(2M ) (x)
This section presents the uniform Haar wavelet and quasilinearization approach based scheme for Burgers’ equation (1) with initial and boundary conditions (2).
where dot and dash denote the differentiation with respect to t and x, respectively, the row vector as is constant in the subinterval t ∈ [ts , ts+1 ].
(11)
i =1
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
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Table 2 Comparison with exact and existing numerical methods of Example 1 for ν = 0.01 at different times t and x. x
t
[19] 1t = 0.0001
[18] 1t = 0.001
[15] 1t = 0.001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.34819 0.27536 0.22752 0.19375 0.07754
0.34244 0.26905 0.22145 0.18813 0.07509
0.34191 0.26896 0.22148 0.18819 0.07511
0.34229 0.26902 ... 0.18817 0.07511
0.34184 0.26891 0.22143 0.18815 0.07510
0.34191 0.26896 0.22148 0.18819 0.07511
0.50
0.4 0.6 0.8 1.0 3.0
0.66543 0.53525 0.44526 0.38047 0.15362
0.67152 0.53406 0.44143 0.37568 0.15020
0.66071 0.52942 0.43914 0.37442 0.15218
0.66797 0.53211 ... 0.37500 0.15018
0.66060 0.52932 0.43905 0.37436 0.15017
0.66071 0.52942 0.43914 0.37442 0.15018
0.75
0.4 0.6 0.8 1.0 3.0
0.91201 0.77132 0.65254 0.56157 0.22874
0.94675 0.78474 0.65659 0.56135 0.22502
0.91027 0.76724 0.64740 0.55605 0.22481
0.93680 0.77724 ... 0.55833 0.22485
0.91026 0.76719 0.64745 0.55608 0.22504
0.91026 0.76724 0.64740 0.55605 0.22481
Table 3 Comparison with exact and existing numerical methods of Example 1 for ν = 0.005, 0.004, 0.003 at different times t and x with 1t = 0.001. x
t
ν = 0.005 [16] 1t = 0.01
Present method
Exact solution
ν = 0.004 Present method
Exact solution
ν = 0.003 Present method
Exact solution
0.25
1 5 10 15
0.18895 0.04696 0.02422 0.01631
0.18874 0.04695 0.02421 0.01631
0.18879 0.04696 0.02422 0.01631
0.18886 0.04696 0.02421 0.01631
0.18889 0.04697 0.02421 0.01631
0.18898 0.04697 0.02422 0.01631
0.18901 0.04698 0.02422 0.01631
0.50
1 5 10 15
0.37837 0.09393 0.04842 0.03244
0.37565 0.09391 0.04842 0.03244
0.37572 0.09392 0.04842 0.03244
0.37591 0.09393 0.04843 0.03259
0.37596 0.09393 0.04843 0.03259
0.37615 0.09394 0.04843 0.03263
0.37619 0.09395 0.04843 0.03263
0.75
1 5 10 15
0.56695 0.14086 0.07112 0.04412
0.55831 0.14083 0.07114 0.04415
0.55838 0.14083 0.07113 0.04413
0.55875 0.14088 0.07221 0.04679
0.55881 0.14089 0.07220 0.04677
0.55919 0.14091 0.07261 0.04840
0.55924 0.14095 0.07260 0.04841
Table 4 Comparison with exact and existing numerical methods of Example 2 for ν = 0.1 at different times t and x. x
t
[19] 1t = 0.0001
[14] 1t = 0.00001
[15] 1t = 0.001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.32091 0.24910 0.20211 0.16782 0.02828
0.32679 0.25117 0.20270 0.16780 0.02804
0.31752 0.24614 0.19955 0.16559 0.02776
0.31743 0.24609 ... 0.16558 0.02776
0.30887 0.24609 0.19952 0.16557 0.02775
0.31752 0.24614 0.19956 0.16560 0.02775
0.50
0.4 0.6 0.8 1.0 3.0
0.58788 0.46174 0.37111 0.30183 0.04185
0.59661 0.46581 0.37293 0.30253 0.04155
0.58454 0.45798 0.36740 0.29834 0.04106
0.58446 0.45791 ... 0.29831 0.04107
0.56979 0.45790 0.36734 0.29829 0.04105
0.58454 0.45798 0.36740 0.29834 0.04106
0.75
0.4 0.6 0.8 1.0 3.0
0.65054 0.50825 0.39068 0.30057 0.03106
0.64680 0.50852 0.39117 0.30066 0.03081
0.64562 0.50268 0.38534 0.29586 0.03044
0.64558 0.50261 ... 0.29582 0.03044
0.62567 0.48715 0.38525 0.29578 0.03043
0.64562 0.50268 0.38534 0.29586 0.03044
Integrating Eq. (11) with respect to t from ts to t, we get the following
By using the boundary conditions (2b), we obtain
u (x, t ) = (t − )
u˙ (0, t ) = f1 (t ),
′′
ts aT(2M )
h(2M ) (x) + u (x, ts ). ′′
(12)
Now, integrating Eq. (12) twice with respect to x from 0 to x, we obtain
u(0, ts ) = f1 (ts ), ′
u(1, ts ) = f2 (ts ) u˙ (1, t ) = f2′ (t ).
(16)
Putting x = 1 in Eqs. (14) and (15) and using the conditions in (16), we obtain u′ (0, t ) − u′ (0, ts ) = −(t − ts )aT(2M ) P2,(2M ) F + f2 (t ) − f1 (t ) − f2 (ts ) + f1 (ts )
(17)
u˙ (0, t ) = f2 (t ) −
(18)
u ( x, t ) = ( t − ) h(2M ) (x) + u (x, ts ) + u′ (0, t ) − u′ (0, ts )
(13)
u(x, t ) = (t − ts )aT(2M ) P2,(2M ) h(2M ) (x) + u(0, t ) + u(x, ts ) − u(0, ts ) + x(u′ (0, t ) − u′ (0, ts ))
(14)
Substituting Eqs. (16)–(18) into Eqs. (13)–(15), and discretizing the results by x → xl , t → ts+1 , we have
u˙ (x, t ) = aT(2M ) P2,(2M ) h(2M ) (x) + u˙ (0, t ) + x u˙ ′ (0, t ).
(15)
u′′ (xl , ts+1 ) = 1t aT(2M ) h(2M ) (xl ) + u′′ (xl , ts ),
′
ts aT(2M ) P1,(2M )
′
′
′
aT(2M ) P2,(2M )
F − f1 (t ). ′
(19)
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Table 5 Comparison with exact and existing numerical methods of Example 2 for ν = 0.01 at different times t and x. x
t
[19] 1t = 0.0001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.36911 0.28905 0.23703 0.20069 0.07865
0.36273 0.28212 ... 0.19467 0.07613
0.36217 0.28197 0.23040 0.19465 0.07613
0.36226 0.28204 0.23045 0.19469 0.07613
0.50
0.4 0.6 0.8 1.0 3.0
0.68818 0.55425 0.46011 0.39206 0.15576
0.69186 0.55125 ... 0.38627 0.15218
0.68357 0.54822 0.45363 0.38561 0.15217
0.68368 0.54832 0.45371 0.38568 0.15218
0.75
0.4 0.6 0.8 1.0 3.0
0.92194 0.78676 0.66777 0.57491 0.23183
0.94940 0.79399 ... 0.57170 0.22778
0.92050 0.78293 0.66264 0.56924 0.22774
0.92050 0.78299 0.66272 0.56932 0.22774
u′ (xl , ts+1 ) = 1t aT(2M ) P1,(2M ) h(2M ) (xl ) − 1t aT(2M ) P2,(2M ) F + f2 (ts+1 ) − f1 (ts+1 ) − f1 (ts ) + f2 (ts ) + u′ (xl , ts )
5. Numerical experiments (20)
u(xl , ts+1 ) = 1t aT(2M ) P2,(2M ) h(2M ) (xl ) + u(xl , ts )
u˙ (xl , ts+1 ) =
− f1 (ts ) + f1 (ts+1 ) + xl (−1t aT(2M ) P2,(2M ) F + f2 (ts+1 ) − f1 (ts+1 ) − f2 (ts ) + f1 (ts ))
(21)
( ) + f1 (ts+1 ) + f2′ (ts+1 ) − f1′ (ts+1 ))
(22)
′
aT(2M ) P2,(2M ) h(2M ) xl xl aT(2M ) P2,(2M ) F
+ (−
where the vector F = [1, 0, . . . , 0] . T
(2M elements)
There are several possibilities for treating the nonlinearity in Eq. (1). But, here the well known technique quasilinearization [39] is used to tackle the nonlinearity in Eq. (1). The nonlinear partial differential equation (1) followed by quasilinearization leads to u˙ (x, ts+1 ) = ν u (x, ts+1 ) − u(x, ts+1 )u (x, ts ) ′′
′
− u(x, ts )u′ (x, ts+1 ) + u(x, ts ) u′ (x, ts ), s = 0, 1, . . . .
(23)
In this section, three test examples are considered to check the efficiency and accuracy of the proposed scheme. In order to measure the accuracy of the numerical scheme error norm L∞ and L2 are calculated. Lagrange’s interpolation is used to find the solution at specified points. The whole computational work has been done with the help of MATLAB software. Example 1. In this example, the author considered Burgers’ equation (1) with initial and boundary conditions in the following form [5,16] u(x, 0) = sin π x,
+ f2′ (ts+1 ) − f1′ (ts+1 ))
A0 +
− f2 (ts+! ) + f1 (ts+! ) + f2 (ts ) − f1 (ts )) u′ (xl , ts ) − 1t aT(2M ) P1,(2M ) h(2M ) (xl )u(xl , ts ) + 1t aT(2M ) P2,(2M ) × F u(xl , ts ) + u(xl , ts )(f1 (ts+! ) + f1 (ts ) (24)
From Eq. (24) the wavelet coefficients aT(2M ) can be successively calculated. This process starts with the following terms
u (xl , 0) = f (xl ).
∞
(28) An exp(−n2 π 2 ν t ) cos(nπ x)
n=1
1
−1 (1 − cos(π x)) dx, 2π ν 0 1 −1 An = 2 exp (1 − cos(π x)) dx. 2π ν 0
+ (f1 (ts ) − f1 (ts+! ))u′ (xl , ts ) + xl (1t aT(2M ) P2,(2M ) F
u′ (xl , 0) = f ′ (xl ),
An exp(−n2 π 2 ν t ) n sin(nπ x)
n =1
u(x, t ) =
× h(2M ) (xl )u′ (xl , ts ) − u′ (xl , ts ) u(xl , ts )
′′
∞
2π ν
A0 =
= ν 1t aT(2M ) h(2M ) (xl ) + ν u′′ (xl , ts ) − 1t aT(2M ) P2,(2M )
′′
(26) (27)
where
aT(2M ) P2,(2M ) h(2M ) (xl ) + f1′ (ts+1 ) + xl (−aT(2M ) P2,(2M ) F
u(xl , 0) = f (xl ),
t > 0.
The exact solution of example is obtained by Hopf–Cole transformation and given by
Now, discretizing the result (23) by x → xl , and using Eqs. (19)– (22), we have
− f2 (ts ) − u′ (xl , ts )) + u(xl , ts )u′ (xl , ts ).
0 ≤ x ≤ 1,
u(0, t ) = u(1, t ) = 0,
(25)
Then, by putting the calculated wavelet coefficients aT(2M ) into Eqs. (19)–(20) we can successively calculate the approximate solutions at different times.
exp
(29)
The numerical solutions of the example are presented for ν = 0.1, 0.01, 0.005, 0.004, 0.003 with 1t = 0.001, M = 64 in Tables 1–3 and Figs. 1–5. The results are compared with [14–16,18,19] and it is found that these are much better than the results presented in [14,18,19]. The figures are depicted upto time t ≤ 3, which exhibit correct physical behavior of the problem. Figs. 3 and 4 show the physical behavior of numerical solutions in 3D and contour which is similar to the 2D figures. Fig. 5 compares the exact and numerical solutions at different times. Example 2. Consider Burgers’ equation (1) with the following initial and boundary conditions [4,13,17] u(x, 0) = 4x(1 − x), u(0, t ) = u(1, t ) = 0,
0 ≤ x ≤ 1, t > 0.
(30) (31)
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
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Table 6 Comparison of exact and numerical solutions of Example 2 for ν = 0.005, 0.004, 0.003 at different times t and x with 1t = 0.001. x
t
ν = 0.005
ν = 0.004
ν = 0.003
Present method
Exact solution
Present method
Exact solution
Present method
Exact solution
0.25
1 5 10 15
0.19604 0.04741 0.02433 0.01636
0.19609 0.04741 0.02434 0.01636
0.19636 0.04744 0.02434 0.01637
0.19641 0.04747 0.02434 0.01637
0.19668 0.04746 0.02434 0.01637
0.19673 0.04748 0.02434 0.01637
0.50
1 5 10 15
0.38795 0.09481 0.04866 0.03255
0.38797 0.09482 0.04866 0.03255
0.38842 0.09491 0.04868 0.03270
0.38846 0.09493 0.04869 0.03270
0.38890 0.09491 0.04870 0.03274
0.38894 0.09494 0.04871 0.03274
0.75
1 5 10 15
0.57248 0.14215 0.07152 0.04433
0.57250 0.14215 0.07151 0.04432
0.57312 0.14224 0.07258 0.04696
0.57315 0.14225 0.07257 0.04695
0.57375 0.14232 0.07298 0.04857
0.57378 0.14234 0.07297 0.04857
Fig. 6. Physical behavior of numerical solutions of Example 2 for ν = 0.1 at different times t with 1t = 0.001.
Fig. 7. Physical behavior of numerical solutions of Example 2 for ν = 0.01 at different times t with 1t = 0.001. Table 7 Absolute error of Example 3 for α = 2, 1t = 0.001 at different times t and v .
v
t = 0.001
t = 0.01
t = 0.1
t = 0.5
1.0 0.5 0.2 0.1 0.01 0.0001 0.00001
5.71475E−06 7.27810E−07 4.71082E−08 5.91078E−09 5.93092E−12 8.13151E−18 4.47233E−19
4.19416E−05 6.20899E−06 4.45567E−07 5.80581E−08 6.03414E−11 8.67802E−17 7.13201E−19
1.29318E−04 3.44228E−05 6.36510E−06 1.40456E−06 2.88883E−09 6.56381E−15 8.46789E−18
1.93815E−04 8.37713E−05 2.47738E−05 8.62965E−06 5.78971E−08 8.86360E−14 1.87445E−16
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Fig. 8. Physical behavior of numerical solutions of Example 2 in 3D and contour for ν = 0.1 at different times t with 1t = 0.001.
Fig. 9. Physical behavior of numerical solutions of Example 2 in 3D and contour for ν = 0.01 at different times t with 1t = 0.001.
Fig. 10. Physical behavior of numerical (left) and exact (right) solutions of Example 2 for ν = 0.01 at different times t with 1t = 0.001. Table 8 Comparison of L∞ and L2 errors with existing numerical methods of Example 3 for ν = 0.01, α = 100, 1t = 0.01 at t = 1.0. N
10 20 40 80
Kaysar [26]
Mittal and Jain [27]
2M
L∞
L2
L∞
L2
4.8808E−07 1.4305E−07 5.6677E−08 3.4992E−08
3.4545E−07 1.0124E−07 4.0028E−08 4.0028E−08
4.6280E−07 1.1640E−07 2.9068E−08 7.2706E−09
3.2840E−07 8.1921E−08 2.0470E−08 5.1194E−09
08 16 32 64
Present method L∞
L2
1.60651E−08 2.12402E−09 1.64798E−09 2.48655E−09
9.90194E−10 1.71750E−10 2.06165E−10 4.44806E−10
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
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Table 9 Comparison of L∞ and L2 errors with existing numerical methods of Example 3 for ν = 0.005, α = 100, 1t = 0.01 at t = 1.0. N
Kaysar [26]
10 20 40 80
Mittal and jain [27]
2M
L∞
L2
L∞
L2
1.2458E−07 3.3944E−08 1.1249E−08 5.5490E−09
8.8189E−08 2.4029E−08 7.9424E−09 3.9178E−09
1.215E−07 3.062E−08 7.644E−09 7.644E−09
8.631E−08 2.153E−08 5.378E−09 1.345E−09
Present method
08 16 32 64
L∞
L2
2.07596E−09 2.73231E−09 2.19142E−10 3.27188E−10
1.27757E−10 2.18337E−11 2.73446E−11 5.85243E−11
Fig. 11. Physical behavior of numerical solutions of Example 3 at t = 0.001, α = 2 (left) and for t ≤ 4 with 1t = 0.0001, ν = 0.01, α = 100 (right).
Fig. 12. Physical behavior of numerical solutions of Example 3 in 3D and contour for different values of ν with 1t = 0.0001 at t = 0.001.
The exact solution of the example is obtained by the Hopf–Cole transformation and is given by ∞
2π ν
An exp(−n2 π 2 ν t ) n sin(nπ x)
n =1
u(x, t ) =
A0 +
∞
(32) An exp(−n2 π 2 ν t ) cos(nπ x)
Example 3. Considering Burgers’ equation (1) with boundary conditions [4,26]
n =1
where 1
−1 2 (3x − 2 x3 ) dx, 3ν 0 1 −1 2 An = exp (3x − 2 x3 ) cos(nπ x)dx. 3ν 0
A0 =
is found that these are much better than the results presented in [14,16,19]. The figures are depicted upto time t ≤ 3, which exhibit correct physical behavior of the problem. Figs. 8 and 9 show the physical behavior of numerical solutions in 3D and contour which is similar to the 2D figures while Fig. 10 compares the exact and numerical solutions at different times.
u(0, t ) = u(1, t ) = 0,
exp
t > 0,
(34)
and with an exact solution (33)
In this example, numerical solutions are presented for ν = 0.1, 0.01, 0.005, 0.004, 0.003 with 1t = 0.001, M = 64 in Tables 4– 6 and Figs. 6–10. The results are compared with [14–16,19] and it
u(x, t ) =
2 2 ν π e−π ν t sin(π x)
σ + e−π 2 ν t cos(π x)
,
0≤x≤1
(35)
where σ > 1 is a parameter. The numerical solutions of the example are presented for ν = 1, 0.5, 0.2, 0.1, 0.01, 0.0001, 0.00001 with 1t = 0.001, M = 64
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Fig. 13. Physical behavior of numerical (left) and exact (right) solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.0001.
in Tables 7–9 and Figs. 11–13. Tables 7–9 show the L2 and L∞ errors at different values of T , ν, α . The results are compared with [4,26,27] and it is found that these are more better than the results presented in [4,26,27]. The errors L2 and L∞ are smaller than the errors presented in [26,27]. The physical behavior of the solution depicted in Fig. 11 is similar as depicted in [26,27]. Fig. 12 shows the physical behavior of numerical solutions in 3D and contour plot while Fig. 13 compares the exact and numerical solutions at different times. 6. Conclusions In this paper, an efficient numerical scheme based on Haar wavelets and the quasilinearization process is developed for solving nonlinear Burgers’ equation with Dirichlet’s boundary conditions. The scheme is tested on three problems and the obtained numerical results are quite satisfactory. The obtained numerical results are compared with the existing numerical and exact solutions. It is concluded that the present technique gives better accuracy in comparison to the other numerical techniques [4,14,16, 18,19,26,27] available in the literature. The main advantage of the Haar wavelet based scheme is that the present scheme is able to capture the behavior of numerical solutions at a small coefficient of kinematic viscosity ν = 0.1, 0.01, 0.005, 0.004, 0.003, where most of the numerical methods fail. The present scheme with some modifications seems to be easily extended to solve model equations including more mechanical, physical or biophysical effects, such as nonlinear convection, reaction, linear diffusion and dispersion. Acknowledgments The author would like to express his thanks to Prof. R.C. Mittal (Head, Department of Mathematics, Indian Institute of Technology Roorkee, India) for his illuminating advice and valuable discussion. Also, the author is very grateful to the reviewers for their valuable suggestions to improve the quality of the paper. References [1] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43 (1915) 163–170. [2] J.M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, North-Holland Pub. Co., 1939, pp. 1–53. [3] J.M. Burgers, A mathematical model illustrating the theory of turbulence, in: Adv. in Appl. Mech., Vol. I, Academic Press, New York, 1948, pp. 171–199. [4] Asai Asaithambi, Numerical solution of the Burgers’ equation by automatic differentiation, Appl. Math. Comput. 216 (2010) 2700–2708.
[5] Abdulkadir Dogan, A Galerkin finite element approach to Burgers’ equation, Appl. Math. Comput. 157 (2004) 331–346. [6] A.H.A. Ali, G.A. Gardner, L.R.T. Gardner, A collocation solution for Burgers’ equation using cubic B-spline finite elements, Comput. Methods Appl. Mech. Eng. 100 (1992) 325–337. [7] A.H. Khater, R.S. Temsah, M.M. Hassan, A Chebyshev spectral collocation method for solving Burgers’ type equations, J. Comput. Appl. Math. 222 (2008) 333–350. [8] A.K. Khalifa, Khalida Inayat Noor, Muhammad Aslam Noor, Some numerical methods for solving Burgers equation, Int. J. Phys. Sci. 6 (7) (2011) 1702–1710. [9] A. Korkmaz, Shock wave simulations using sinc differential quadrature method, Eng. Comput. 28 (6) (2011) 654–674. [10] A. Korkmaz, Idris Dag, Polynomial based differential quadrature method for numerical solution of nonlinear Burgers’ equation, J. Franklin Inst. 348 (10) (2011) 2863–2875. [11] A. Korkmaz, A. Murat Aksoy, Idris Dag, Quartic B-spline differential quadrature method, Int. J. Nonlinear Sci. 11 (4) (2011) 403–411. [12] B. Saka, I. Dag, Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation, Chaos Solitons Fractals 32 (2007) 1125–1137. [13] E.N. Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers’ equation, Appl. Math. Comput. 174 (2006) 884–896. [14] T. Ozis, E.N. Aksan, A. Ozdes, A finite element approach for solution of Burgers’ equation, Appl. Math. Comput. 139 (2003) 417–428. [15] I.A. Hassanien, A.A. Salama, H.A. Hosham, Fourth-order finite difference method for solving Burgers’ equation, Appl. Math. Comput. 170 (2005) 781–800. [16] M.K. Kadalbajoo, K.K. Sharma, A. Awasthi, A parameter-uniform implicit difference scheme for solving time dependent Burgers’ equation, Appl. Math. Comput. 170 (2005) 1365–1393. [17] Min Xu, Ren-Hong Wang, Ji-Hong Zhang, Qin Fang, A novel numerical scheme for solving Burgers’ equation, Appl. Math. Comput. 217 (2011) 4473–4482. [18] S. Kutulay, A.R. Bahadir, A. Odes, Numerical solution of the one-dimensional Burgers’ equation: explicit and exact-explicit finite difference methods, J. Comput. Appl. Math. 103 (1999) 251–261. [19] S. Kutulay, A. Esen, I. Dag, Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004) 21–33. [20] W. Liao, An implicit fourth-order compact finite difference scheme for onedimensional Burgers’ equation, Appl. Math. Comput. 206 (2008) 755–764. [21] A.H.A.E. Tabatabaei, E. Shakour, M. Dehghan, Some implicit methods for the numerical solution of Burgers equation, Appl. Math. Comput. 191 (2007) 560–570. [22] M. Dehghan, A. Hamidi, M. Shakourifar, The solution of coupled Burgers equations using Adomian–Pade technique, Appl. Math. Comput. 189 (2007) 1034–1047. [23] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burgers and coupled Burgers equations, J. Comput. Appl. Math. 181 (2) (2005) 245–251. [24] M.M. Rashidi, G. Domairry, S. Dinarvand, Approximate solutions for the Burgers’ and regularized long wave equations by means of the homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 708–717. [25] M.M. Rashidi, E. Erfani, New analytic method for solving Burgers’ and nonlinear heat transfer equations and comparison with HAM, Comput. Phys. Comm. 180 (2009) 1539–1544. [26] Kaysar Rahman, Nurmamat Helil, Rahmatjan Yimin, Some new semiimplicit finite difference schemes for numerical solution of Burgers equation, International Conference on Computer Application and System Modeling (ICCASM 2010) 978-1-4244-7237-6/10/$26.00 @ 2010 IEEE V14–451.
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423 [27] R.C. Mittal, R.K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method, Appl. Math. Comput. 218 (2012) 7839–7855. [28] S. Bertoluzza, G. Naldi, J.C. Ravel, Wavelet methods for the numerical solution of boundary value problems on the interval, in: K. Chui, L. Montefusco, L. Puccio (Eds.), Wavelets: Theory, Algorithms and Applications, Academic Press, 1994, pp. 425–428. [29] V. Comincioli, G. Naldi, T. Scapolla, A wavelet-based method for numerical solution of nonlinear evolution equations, Appl. Numer. Math. 33 (2000) 291–297. [30] M.Q. Chen, C. Hwang, Y.P. Shih, The computation of wavelet-Galerkin approximation on a bounded interval, Int. J. Numer. Methods Eng. 39 (1996) 2921–2944. [31] A. Avudainayagam, C. Vani, Wavelet-Galerkin solutions of quasilinear hyperbolic conservation equations, Commun. Numer. Methods Eng. 15 (1999) 589–601. [32] U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68 (2005) 127–143.
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Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc
A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation Ram Jiwari ∗ School of Mathematics & Computer Applications, Thapar University, Patiala 147004, India
article
abstract
info
Article history: Received 19 April 2012 Received in revised form 31 May 2012 Accepted 16 June 2012 Available online 19 June 2012
In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs. © 2012 Elsevier B.V. All rights reserved.
Keywords: Haar wavelets Burgers’ equation Multi-resolution analysis Collocation points
1. Introduction Consider the one-dimensional Burgers’ equation
∂u ∂ 2u ∂u −ν 2 +u = 0, ∂t ∂x ∂x
(x, t ) ∈ Ω × (0, T ],
(1)
with the initial condition u(x, 0) = f (x),
0 ≤ x ≤ 1,
(2a)
and the boundary conditions u(0, t ) = f1 (t ),
u(1, t ) = f2 (t ) 0 ≤ t ≤ T
(2b)
where Ω = (0, 1), ν > 0 is the coefficient of kinematic viscosity and the prescribed function f (x) is sufficiently smooth. The nonlinear homogeneous quasilinear parabolic partial differential equation is the simplest nonlinear model equation for diffusive waves in fluid dynamics. Burgers’ equation arises in many physical problems including one-dimensional turbulence, sound waves in a viscous medium, shock waves in a viscous medium, waves in fluid filled viscous elastic tubes, and magnetohydrodynamic waves in a medium with finite electrical conductivity. Such type of equation was first introduced by Bateman [1] in 1915 and he proposed the steady-state solution of the problem. In 1948, Burgers [2,3] introduced this equation to capture some features of turbulent fluid in a channel caused by the interaction of the
∗
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0010-4655/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2012.06.009
opposite effects of convection and diffusion. Therefore, Eq. (1) is referred to as ‘‘Burgers’ equation’’. The structure of Burgers’ equation is roughly similar to that of Navier–Stoke’s equations due to the presence of the nonlinear convection term and the occurrence of the diffusion term with viscosity coefficient. So this equation can be considered as a simplified form of Navier–Stoke’s equations. The study of the general properties of Burgers’ equation has attracted attention of scientific community due to its applications in various fields such as gas dynamics, heat conduction, elasticity, etc. The study of the solution of Burgers’ equation has been carried out for last half Century and still it is an active area of research to develop some better numerical scheme to approximate its solution. So far various numerical algorithms such as the automatic differentiation method [4], Galerkin finite element method [5], cubic B-splines collocation method [6], spectral collocation method [7,8], Sinc Differential Quadrature Method [9], Polynomial based differential quadrature method [10], Quartic B-splines Differential Quadrature Method [11], Quartic B-splines collocation method [12], Quadratic B-splines finite element method [13], finite element method [14], Fourth-order finite difference method [15], a parameter-uniform implicit difference scheme [16], A novel numerical scheme [17], explicit and exactexplicit finite difference methods [18], least-squares quadratic B-splines finite element method [19], implicit fourth-order compact finite difference scheme [20], some implicit methods [21], Adomian–Pade technique [22], variational iteration method [23], homotopy analysis method [24], differential transform method and the homotopy analysis method [25], Semi-Implicit Finite Difference Schemes [26], modified cubic B-splines collocation method [27] etc.
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Wavelets have been used for the solution of partial differential equations (PDEs) since the 1980s. The good features of this approach are possibility to detect singularities, irregular structure and transient phenomena exhibited by the analyzed equations. Most of the wavelets algorithms for solving PDE are based on the collocation [28,29] or the Galerkin techniques method [28,30,31]. Evidently, all attempts to simplify the wavelet solutions for PDE are welcome; one of the possibilities for this is to make use of the Haar wavelet family. Haar wavelets (which are Daubechies of order 1) consist of piecewise constant functions and are therefore the simplest orthonormal wavelets with a compact support. The Haar wavelets have gained popularity among researchers for their useful properties such as simple applicability, orthogonality and compact support. Compact support of the Haar wavelet basis permits straight inclusion of the different types of boundary conditions in the numeric algorithms. A drawback of the Haar wavelets is their discontinuity. Since the derivatives do not exist in the breaking points it is not possible to apply the Haar wavelets for solving PDE directly. Lepik [32–34] had solved higher order as well as nonlinear ODEs by using the Haar wavelet method. Recently, Hariharan and his associates [35,36] have used Haar wavelets methods for the solution of some nonlinear PDEs. In this paper, an efficient numerical scheme based on uniform Haar wavelets and quasilinearization process is proposed for the numerical simulation of the nonlinear homogeneous quasilinear parabolic Burger’s equation. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents solutions of boundary value problems. The accuracy of the proposed scheme is demonstrated by three test problems. The numerical results are compared with existing numerical and exact solutions and it is found that the proposed scheme produce better results. The use of uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.
Wavelet transform or wavelet analysis is a recently developed mathematical tool for many problems. One of the popular families of wavelet is Haar wavelets. The Haar function is in fact the Daubechies wavelet of order 1. Due to its simplicity, the Haar wavelet had become an effective tool for solving many problems arising in many branches of sciences. Haar functions have been used since 1910. It was introduced by the Hungarian mathematician Alfred Haar. The Haar function is an odd rectangular pulse pair, is the simplest and oldest orthonormal wavelet with compact support. There are different definitions for the Haar function and various generalizations have been used and published. Haar showed that certain square wave function could be translated and scaled to create a basis set that span L2 . Year later, it was seen that the system of Haar is a particular wavelet system. If we choose scaling function to have compact support over 0 ≤ x ≤ 1, that is the Haar wavelet family for x ∈ [0, 1) is defined as hi ( x ) =
1
−1 0
h1 (x) =
x ∈ [0, 1), otherwise.
1 0
(5)
If we want to solve partial differential equations of any order, we need the following integrals pi,1 (x) =
x
hi (x′ )dx′ , 0
(6)
x
pi,v+1 (x) =
pi,v (x′ )dx′ ,
v = 1, 2, 3, . . . .
0
Taking into account (3) the integrals (6) can be calculated analytically; by doing so we obtained the following x − ξ1 ξ3 − x 0
pi,1 (x) =
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ), elsewhere
(x − ξ1 )2 2 1 (ξ3 − x)2 pi,2 (x) = 4m2 − 2 1 4m2 0
(7)
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ),
(8)
x ∈ [ξ1 , 1) elsewhere.
3. Function approximation Any function y(x) which is square integrable in the interval (0, 1) can be expressed in the following form of Haar wavelets y(x) =
∞
ai hi (x).
(9)
i=1
2. Haar wavelets
index i in Eq. (4) is calculated from the formula i = m + k + 1. In the case of minimal values m = 1, k = 0, we have i = 2. The maximum value of i is i = 2M = 2J +1 . For i = 1, the function h1 (x) is the scaling function for the family of the Haar wavelets which is defined as
x ∈ [ξ1 , ξ2 ), x ∈ [ξ2 , ξ3 ), otherwise
(3)
The above series terminates at finite terms if y(x) is piecewise constant or can be approximated as piecewise constant during each subinterval, then y(x) will be terminated at finite terms, that is y(x) =
2M
ai hi (x) = aT(2M ) h(2M ) (x),
(10)
i=1
where the coefficients aT(2M ) and the Haar function vector h(2M ) (x) are defined as aT(2M ) = [a1 , a2 , . . . , a2M ]
and
h(2M ) (x) = [h(1) (x), h(2) (x), . . . , h(2M ) (x)]T where T denotes the transpose and M = 2j . The best way to understand wavelets is through a multiresolution analysis. Given a function y(x) ∈ L2 (R) a multi-resolution analysis (MRA) of L2 (R) produces a sequence of subspaces Uj , Uj+1 , . . . such that the projections of y(x) onto these spaces give finer and finer approximations of the function y(x) as j → ∞. The details of MRA is as follows. 3.1. Multi-resolution analysis
where (4)
A multi-resolution analysis of L2 (R) is defined as a sequence of closed subspaces Uj ⊂ L2 (R), j ∈ Z with the following properties.
In the above definition the integer m = 2j , j = 0, 1, . . . , J indicates the level of the wavelet and the integer k = 0, 1, . . . , m − 1 is the translation parameter. The maximal level of resolution is J. The
(i) · · · ⊂ U−1 ⊂ U0 ⊂ U1 ⊂ · · ·. 2 (ii) The space Uj satisfying j∈Z Uj is dense in L (R) and j∈Z Uj = 0.
ξ1 =
k m
,
ξ2 =
k + 0.5 m
,
ξ3 =
k+1 m
.
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
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Fig. 1. Physical behavior of numerical solutions of Example 1 for ν = 0.1 at different times t with 1t = 0.001.
Fig. 2. Physical behavior of numerical solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.001.
Fig. 3. Physical behavior of numerical solutions of Example 1 in 3D and contour for ν = 0.1 at different times t with 1t = 0.001.
(iii) If g (x) ∈ U0 , g (2j x) ∈ Uj i.e. the space Uj is a scaled version of the central space U0 . (iv) If g (x) ∈ U0 , g (2j x − k) ∈ Uj i.e. all the Uj are invariant under translation. (v) There exists Φ ∈ U0 such that Φ (x − k); k ∈ Z is a Riesz basis in U0 . The space Uj is used to approximate general functions by defining appropriate projection of these functions onto these spaces. Since
the union of all the Uj is dense in L2 (R), it guarantees that any function in L2 (R) can be approximated arbitrarily close by such projections. As an example the space Uj can be defined like J +1
Uj = Wj−1 ⊕ Uj−1 = Wj−1 ⊕ Wj−2 ⊕ Vj−2 = · · · = ⊕ Wj ⊕ V0 j =1
then the scaling function h1 (x) generates an MRA for the sequence of spaces { Uj , j ∈ Z } by translation and dilation as defined in Eqs. (3) and (5). For each j the space Wj serves as the orthogonal
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Fig. 4. Physical behavior of numerical solutions of Example 1 in 3D and contour for ν = 0.01 at different times t with 1t = 0.001.
Fig. 5. Physical behavior of numerical (left) and exact (right) solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.001. Table 1 Comparison with exact and existing numerical methods of Example 1 for ν = 0.1 at different times t and x. x
t
[18] 1t = 0.001
[14]
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.30891 0.24075 0.19568 0.16257 0.02720
0.31429 0.24373 0.19758 0.16391 0.02743
0.30881 0.24069 ... 0.16254 0.02720
0.30887 0.24070 0.19566 0.16255 0.02721
0.30889 0.24074 0.19568 0.16256 0.02720
0.50
0.4 0.6 0.8 1.0 3.0
0.56964 0.44721 0.35924 0.29192 0.04021
0.57636 0.45169 0.36245 0.29437 0.04057
0.56955 0.44714 ... 0.29188 0.04021
0.56956 0.44715 0.35920 0.29188 0.04022
0.56963 0.44721 0.35924 0.29192 0.04021
0.75
0.4 0.6 0.8 1.0 3.0
0.62542 0.48721 0.37392 0.28748 0.02977
0.62592 0.49034 0.37713 0.29016 0.01334
0.62540 0.48715 ... 0.28744 0.02978
0.62540 0.48716 0.37389 0.28743 0.02978
0.62544 0.48721 0.37392 0.28747 0.02977
complement of Uj in Uj+1 . The space Wj includes all the functions in Uj+1 that are orthogonal to all those in Uj under some chosen inner product. The set of functions which form basis for the space Wj are called wavelets [37,38].
Let us divide the interval (0, 1] into N equal parts of length 1t = 1/N and denote ts = (s − 1)1t , s = 1, 2, . . . , N. We assume the following for the solution of Eq. (1), assumed by Lepik [33] 2M
4. Uniform Haar wavelet based scheme for Burgers’ equation
u˙ ′′ (x, t ) =
as (i)hi (x) = aT(2M ) h(2M ) (x)
This section presents the uniform Haar wavelet and quasilinearization approach based scheme for Burgers’ equation (1) with initial and boundary conditions (2).
where dot and dash denote the differentiation with respect to t and x, respectively, the row vector as is constant in the subinterval t ∈ [ts , ts+1 ].
(11)
i =1
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
2417
Table 2 Comparison with exact and existing numerical methods of Example 1 for ν = 0.01 at different times t and x. x
t
[19] 1t = 0.0001
[18] 1t = 0.001
[15] 1t = 0.001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.34819 0.27536 0.22752 0.19375 0.07754
0.34244 0.26905 0.22145 0.18813 0.07509
0.34191 0.26896 0.22148 0.18819 0.07511
0.34229 0.26902 ... 0.18817 0.07511
0.34184 0.26891 0.22143 0.18815 0.07510
0.34191 0.26896 0.22148 0.18819 0.07511
0.50
0.4 0.6 0.8 1.0 3.0
0.66543 0.53525 0.44526 0.38047 0.15362
0.67152 0.53406 0.44143 0.37568 0.15020
0.66071 0.52942 0.43914 0.37442 0.15218
0.66797 0.53211 ... 0.37500 0.15018
0.66060 0.52932 0.43905 0.37436 0.15017
0.66071 0.52942 0.43914 0.37442 0.15018
0.75
0.4 0.6 0.8 1.0 3.0
0.91201 0.77132 0.65254 0.56157 0.22874
0.94675 0.78474 0.65659 0.56135 0.22502
0.91027 0.76724 0.64740 0.55605 0.22481
0.93680 0.77724 ... 0.55833 0.22485
0.91026 0.76719 0.64745 0.55608 0.22504
0.91026 0.76724 0.64740 0.55605 0.22481
Table 3 Comparison with exact and existing numerical methods of Example 1 for ν = 0.005, 0.004, 0.003 at different times t and x with 1t = 0.001. x
t
ν = 0.005 [16] 1t = 0.01
Present method
Exact solution
ν = 0.004 Present method
Exact solution
ν = 0.003 Present method
Exact solution
0.25
1 5 10 15
0.18895 0.04696 0.02422 0.01631
0.18874 0.04695 0.02421 0.01631
0.18879 0.04696 0.02422 0.01631
0.18886 0.04696 0.02421 0.01631
0.18889 0.04697 0.02421 0.01631
0.18898 0.04697 0.02422 0.01631
0.18901 0.04698 0.02422 0.01631
0.50
1 5 10 15
0.37837 0.09393 0.04842 0.03244
0.37565 0.09391 0.04842 0.03244
0.37572 0.09392 0.04842 0.03244
0.37591 0.09393 0.04843 0.03259
0.37596 0.09393 0.04843 0.03259
0.37615 0.09394 0.04843 0.03263
0.37619 0.09395 0.04843 0.03263
0.75
1 5 10 15
0.56695 0.14086 0.07112 0.04412
0.55831 0.14083 0.07114 0.04415
0.55838 0.14083 0.07113 0.04413
0.55875 0.14088 0.07221 0.04679
0.55881 0.14089 0.07220 0.04677
0.55919 0.14091 0.07261 0.04840
0.55924 0.14095 0.07260 0.04841
Table 4 Comparison with exact and existing numerical methods of Example 2 for ν = 0.1 at different times t and x. x
t
[19] 1t = 0.0001
[14] 1t = 0.00001
[15] 1t = 0.001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.32091 0.24910 0.20211 0.16782 0.02828
0.32679 0.25117 0.20270 0.16780 0.02804
0.31752 0.24614 0.19955 0.16559 0.02776
0.31743 0.24609 ... 0.16558 0.02776
0.30887 0.24609 0.19952 0.16557 0.02775
0.31752 0.24614 0.19956 0.16560 0.02775
0.50
0.4 0.6 0.8 1.0 3.0
0.58788 0.46174 0.37111 0.30183 0.04185
0.59661 0.46581 0.37293 0.30253 0.04155
0.58454 0.45798 0.36740 0.29834 0.04106
0.58446 0.45791 ... 0.29831 0.04107
0.56979 0.45790 0.36734 0.29829 0.04105
0.58454 0.45798 0.36740 0.29834 0.04106
0.75
0.4 0.6 0.8 1.0 3.0
0.65054 0.50825 0.39068 0.30057 0.03106
0.64680 0.50852 0.39117 0.30066 0.03081
0.64562 0.50268 0.38534 0.29586 0.03044
0.64558 0.50261 ... 0.29582 0.03044
0.62567 0.48715 0.38525 0.29578 0.03043
0.64562 0.50268 0.38534 0.29586 0.03044
Integrating Eq. (11) with respect to t from ts to t, we get the following
By using the boundary conditions (2b), we obtain
u (x, t ) = (t − )
u˙ (0, t ) = f1 (t ),
′′
ts aT(2M )
h(2M ) (x) + u (x, ts ). ′′
(12)
Now, integrating Eq. (12) twice with respect to x from 0 to x, we obtain
u(0, ts ) = f1 (ts ), ′
u(1, ts ) = f2 (ts ) u˙ (1, t ) = f2′ (t ).
(16)
Putting x = 1 in Eqs. (14) and (15) and using the conditions in (16), we obtain u′ (0, t ) − u′ (0, ts ) = −(t − ts )aT(2M ) P2,(2M ) F + f2 (t ) − f1 (t ) − f2 (ts ) + f1 (ts )
(17)
u˙ (0, t ) = f2 (t ) −
(18)
u ( x, t ) = ( t − ) h(2M ) (x) + u (x, ts ) + u′ (0, t ) − u′ (0, ts )
(13)
u(x, t ) = (t − ts )aT(2M ) P2,(2M ) h(2M ) (x) + u(0, t ) + u(x, ts ) − u(0, ts ) + x(u′ (0, t ) − u′ (0, ts ))
(14)
Substituting Eqs. (16)–(18) into Eqs. (13)–(15), and discretizing the results by x → xl , t → ts+1 , we have
u˙ (x, t ) = aT(2M ) P2,(2M ) h(2M ) (x) + u˙ (0, t ) + x u˙ ′ (0, t ).
(15)
u′′ (xl , ts+1 ) = 1t aT(2M ) h(2M ) (xl ) + u′′ (xl , ts ),
′
ts aT(2M ) P1,(2M )
′
′
′
aT(2M ) P2,(2M )
F − f1 (t ). ′
(19)
2418
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
Table 5 Comparison with exact and existing numerical methods of Example 2 for ν = 0.01 at different times t and x. x
t
[19] 1t = 0.0001
[16] 1t = 0.01
Present method 1t = 0.001
Exact solution
0.25
0.4 0.6 0.8 1.0 3.0
0.36911 0.28905 0.23703 0.20069 0.07865
0.36273 0.28212 ... 0.19467 0.07613
0.36217 0.28197 0.23040 0.19465 0.07613
0.36226 0.28204 0.23045 0.19469 0.07613
0.50
0.4 0.6 0.8 1.0 3.0
0.68818 0.55425 0.46011 0.39206 0.15576
0.69186 0.55125 ... 0.38627 0.15218
0.68357 0.54822 0.45363 0.38561 0.15217
0.68368 0.54832 0.45371 0.38568 0.15218
0.75
0.4 0.6 0.8 1.0 3.0
0.92194 0.78676 0.66777 0.57491 0.23183
0.94940 0.79399 ... 0.57170 0.22778
0.92050 0.78293 0.66264 0.56924 0.22774
0.92050 0.78299 0.66272 0.56932 0.22774
u′ (xl , ts+1 ) = 1t aT(2M ) P1,(2M ) h(2M ) (xl ) − 1t aT(2M ) P2,(2M ) F + f2 (ts+1 ) − f1 (ts+1 ) − f1 (ts ) + f2 (ts ) + u′ (xl , ts )
5. Numerical experiments (20)
u(xl , ts+1 ) = 1t aT(2M ) P2,(2M ) h(2M ) (xl ) + u(xl , ts )
u˙ (xl , ts+1 ) =
− f1 (ts ) + f1 (ts+1 ) + xl (−1t aT(2M ) P2,(2M ) F + f2 (ts+1 ) − f1 (ts+1 ) − f2 (ts ) + f1 (ts ))
(21)
( ) + f1 (ts+1 ) + f2′ (ts+1 ) − f1′ (ts+1 ))
(22)
′
aT(2M ) P2,(2M ) h(2M ) xl xl aT(2M ) P2,(2M ) F
+ (−
where the vector F = [1, 0, . . . , 0] . T
(2M elements)
There are several possibilities for treating the nonlinearity in Eq. (1). But, here the well known technique quasilinearization [39] is used to tackle the nonlinearity in Eq. (1). The nonlinear partial differential equation (1) followed by quasilinearization leads to u˙ (x, ts+1 ) = ν u (x, ts+1 ) − u(x, ts+1 )u (x, ts ) ′′
′
− u(x, ts )u′ (x, ts+1 ) + u(x, ts ) u′ (x, ts ), s = 0, 1, . . . .
(23)
In this section, three test examples are considered to check the efficiency and accuracy of the proposed scheme. In order to measure the accuracy of the numerical scheme error norm L∞ and L2 are calculated. Lagrange’s interpolation is used to find the solution at specified points. The whole computational work has been done with the help of MATLAB software. Example 1. In this example, the author considered Burgers’ equation (1) with initial and boundary conditions in the following form [5,16] u(x, 0) = sin π x,
+ f2′ (ts+1 ) − f1′ (ts+1 ))
A0 +
− f2 (ts+! ) + f1 (ts+! ) + f2 (ts ) − f1 (ts )) u′ (xl , ts ) − 1t aT(2M ) P1,(2M ) h(2M ) (xl )u(xl , ts ) + 1t aT(2M ) P2,(2M ) × F u(xl , ts ) + u(xl , ts )(f1 (ts+! ) + f1 (ts ) (24)
From Eq. (24) the wavelet coefficients aT(2M ) can be successively calculated. This process starts with the following terms
u (xl , 0) = f (xl ).
∞
(28) An exp(−n2 π 2 ν t ) cos(nπ x)
n=1
1
−1 (1 − cos(π x)) dx, 2π ν 0 1 −1 An = 2 exp (1 − cos(π x)) dx. 2π ν 0
+ (f1 (ts ) − f1 (ts+! ))u′ (xl , ts ) + xl (1t aT(2M ) P2,(2M ) F
u′ (xl , 0) = f ′ (xl ),
An exp(−n2 π 2 ν t ) n sin(nπ x)
n =1
u(x, t ) =
× h(2M ) (xl )u′ (xl , ts ) − u′ (xl , ts ) u(xl , ts )
′′
∞
2π ν
A0 =
= ν 1t aT(2M ) h(2M ) (xl ) + ν u′′ (xl , ts ) − 1t aT(2M ) P2,(2M )
′′
(26) (27)
where
aT(2M ) P2,(2M ) h(2M ) (xl ) + f1′ (ts+1 ) + xl (−aT(2M ) P2,(2M ) F
u(xl , 0) = f (xl ),
t > 0.
The exact solution of example is obtained by Hopf–Cole transformation and given by
Now, discretizing the result (23) by x → xl , and using Eqs. (19)– (22), we have
− f2 (ts ) − u′ (xl , ts )) + u(xl , ts )u′ (xl , ts ).
0 ≤ x ≤ 1,
u(0, t ) = u(1, t ) = 0,
(25)
Then, by putting the calculated wavelet coefficients aT(2M ) into Eqs. (19)–(20) we can successively calculate the approximate solutions at different times.
exp
(29)
The numerical solutions of the example are presented for ν = 0.1, 0.01, 0.005, 0.004, 0.003 with 1t = 0.001, M = 64 in Tables 1–3 and Figs. 1–5. The results are compared with [14–16,18,19] and it is found that these are much better than the results presented in [14,18,19]. The figures are depicted upto time t ≤ 3, which exhibit correct physical behavior of the problem. Figs. 3 and 4 show the physical behavior of numerical solutions in 3D and contour which is similar to the 2D figures. Fig. 5 compares the exact and numerical solutions at different times. Example 2. Consider Burgers’ equation (1) with the following initial and boundary conditions [4,13,17] u(x, 0) = 4x(1 − x), u(0, t ) = u(1, t ) = 0,
0 ≤ x ≤ 1, t > 0.
(30) (31)
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
2419
Table 6 Comparison of exact and numerical solutions of Example 2 for ν = 0.005, 0.004, 0.003 at different times t and x with 1t = 0.001. x
t
ν = 0.005
ν = 0.004
ν = 0.003
Present method
Exact solution
Present method
Exact solution
Present method
Exact solution
0.25
1 5 10 15
0.19604 0.04741 0.02433 0.01636
0.19609 0.04741 0.02434 0.01636
0.19636 0.04744 0.02434 0.01637
0.19641 0.04747 0.02434 0.01637
0.19668 0.04746 0.02434 0.01637
0.19673 0.04748 0.02434 0.01637
0.50
1 5 10 15
0.38795 0.09481 0.04866 0.03255
0.38797 0.09482 0.04866 0.03255
0.38842 0.09491 0.04868 0.03270
0.38846 0.09493 0.04869 0.03270
0.38890 0.09491 0.04870 0.03274
0.38894 0.09494 0.04871 0.03274
0.75
1 5 10 15
0.57248 0.14215 0.07152 0.04433
0.57250 0.14215 0.07151 0.04432
0.57312 0.14224 0.07258 0.04696
0.57315 0.14225 0.07257 0.04695
0.57375 0.14232 0.07298 0.04857
0.57378 0.14234 0.07297 0.04857
Fig. 6. Physical behavior of numerical solutions of Example 2 for ν = 0.1 at different times t with 1t = 0.001.
Fig. 7. Physical behavior of numerical solutions of Example 2 for ν = 0.01 at different times t with 1t = 0.001. Table 7 Absolute error of Example 3 for α = 2, 1t = 0.001 at different times t and v .
v
t = 0.001
t = 0.01
t = 0.1
t = 0.5
1.0 0.5 0.2 0.1 0.01 0.0001 0.00001
5.71475E−06 7.27810E−07 4.71082E−08 5.91078E−09 5.93092E−12 8.13151E−18 4.47233E−19
4.19416E−05 6.20899E−06 4.45567E−07 5.80581E−08 6.03414E−11 8.67802E−17 7.13201E−19
1.29318E−04 3.44228E−05 6.36510E−06 1.40456E−06 2.88883E−09 6.56381E−15 8.46789E−18
1.93815E−04 8.37713E−05 2.47738E−05 8.62965E−06 5.78971E−08 8.86360E−14 1.87445E−16
2420
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Fig. 8. Physical behavior of numerical solutions of Example 2 in 3D and contour for ν = 0.1 at different times t with 1t = 0.001.
Fig. 9. Physical behavior of numerical solutions of Example 2 in 3D and contour for ν = 0.01 at different times t with 1t = 0.001.
Fig. 10. Physical behavior of numerical (left) and exact (right) solutions of Example 2 for ν = 0.01 at different times t with 1t = 0.001. Table 8 Comparison of L∞ and L2 errors with existing numerical methods of Example 3 for ν = 0.01, α = 100, 1t = 0.01 at t = 1.0. N
10 20 40 80
Kaysar [26]
Mittal and Jain [27]
2M
L∞
L2
L∞
L2
4.8808E−07 1.4305E−07 5.6677E−08 3.4992E−08
3.4545E−07 1.0124E−07 4.0028E−08 4.0028E−08
4.6280E−07 1.1640E−07 2.9068E−08 7.2706E−09
3.2840E−07 8.1921E−08 2.0470E−08 5.1194E−09
08 16 32 64
Present method L∞
L2
1.60651E−08 2.12402E−09 1.64798E−09 2.48655E−09
9.90194E−10 1.71750E−10 2.06165E−10 4.44806E−10
R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
2421
Table 9 Comparison of L∞ and L2 errors with existing numerical methods of Example 3 for ν = 0.005, α = 100, 1t = 0.01 at t = 1.0. N
Kaysar [26]
10 20 40 80
Mittal and jain [27]
2M
L∞
L2
L∞
L2
1.2458E−07 3.3944E−08 1.1249E−08 5.5490E−09
8.8189E−08 2.4029E−08 7.9424E−09 3.9178E−09
1.215E−07 3.062E−08 7.644E−09 7.644E−09
8.631E−08 2.153E−08 5.378E−09 1.345E−09
Present method
08 16 32 64
L∞
L2
2.07596E−09 2.73231E−09 2.19142E−10 3.27188E−10
1.27757E−10 2.18337E−11 2.73446E−11 5.85243E−11
Fig. 11. Physical behavior of numerical solutions of Example 3 at t = 0.001, α = 2 (left) and for t ≤ 4 with 1t = 0.0001, ν = 0.01, α = 100 (right).
Fig. 12. Physical behavior of numerical solutions of Example 3 in 3D and contour for different values of ν with 1t = 0.0001 at t = 0.001.
The exact solution of the example is obtained by the Hopf–Cole transformation and is given by ∞
2π ν
An exp(−n2 π 2 ν t ) n sin(nπ x)
n =1
u(x, t ) =
A0 +
∞
(32) An exp(−n2 π 2 ν t ) cos(nπ x)
Example 3. Considering Burgers’ equation (1) with boundary conditions [4,26]
n =1
where 1
−1 2 (3x − 2 x3 ) dx, 3ν 0 1 −1 2 An = exp (3x − 2 x3 ) cos(nπ x)dx. 3ν 0
A0 =
is found that these are much better than the results presented in [14,16,19]. The figures are depicted upto time t ≤ 3, which exhibit correct physical behavior of the problem. Figs. 8 and 9 show the physical behavior of numerical solutions in 3D and contour which is similar to the 2D figures while Fig. 10 compares the exact and numerical solutions at different times.
u(0, t ) = u(1, t ) = 0,
exp
t > 0,
(34)
and with an exact solution (33)
In this example, numerical solutions are presented for ν = 0.1, 0.01, 0.005, 0.004, 0.003 with 1t = 0.001, M = 64 in Tables 4– 6 and Figs. 6–10. The results are compared with [14–16,19] and it
u(x, t ) =
2 2 ν π e−π ν t sin(π x)
σ + e−π 2 ν t cos(π x)
,
0≤x≤1
(35)
where σ > 1 is a parameter. The numerical solutions of the example are presented for ν = 1, 0.5, 0.2, 0.1, 0.01, 0.0001, 0.00001 with 1t = 0.001, M = 64
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R. Jiwari / Computer Physics Communications 183 (2012) 2413–2423
Fig. 13. Physical behavior of numerical (left) and exact (right) solutions of Example 1 for ν = 0.01 at different times t with 1t = 0.0001.
in Tables 7–9 and Figs. 11–13. Tables 7–9 show the L2 and L∞ errors at different values of T , ν, α . The results are compared with [4,26,27] and it is found that these are more better than the results presented in [4,26,27]. The errors L2 and L∞ are smaller than the errors presented in [26,27]. The physical behavior of the solution depicted in Fig. 11 is similar as depicted in [26,27]. Fig. 12 shows the physical behavior of numerical solutions in 3D and contour plot while Fig. 13 compares the exact and numerical solutions at different times. 6. Conclusions In this paper, an efficient numerical scheme based on Haar wavelets and the quasilinearization process is developed for solving nonlinear Burgers’ equation with Dirichlet’s boundary conditions. The scheme is tested on three problems and the obtained numerical results are quite satisfactory. The obtained numerical results are compared with the existing numerical and exact solutions. It is concluded that the present technique gives better accuracy in comparison to the other numerical techniques [4,14,16, 18,19,26,27] available in the literature. The main advantage of the Haar wavelet based scheme is that the present scheme is able to capture the behavior of numerical solutions at a small coefficient of kinematic viscosity ν = 0.1, 0.01, 0.005, 0.004, 0.003, where most of the numerical methods fail. The present scheme with some modifications seems to be easily extended to solve model equations including more mechanical, physical or biophysical effects, such as nonlinear convection, reaction, linear diffusion and dispersion. Acknowledgments The author would like to express his thanks to Prof. R.C. Mittal (Head, Department of Mathematics, Indian Institute of Technology Roorkee, India) for his illuminating advice and valuable discussion. Also, the author is very grateful to the reviewers for their valuable suggestions to improve the quality of the paper. References [1] H. Bateman, Some recent researches on the motion of fluids, Mon. Weather Rev. 43 (1915) 163–170. [2] J.M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, North-Holland Pub. Co., 1939, pp. 1–53. [3] J.M. Burgers, A mathematical model illustrating the theory of turbulence, in: Adv. in Appl. Mech., Vol. I, Academic Press, New York, 1948, pp. 171–199. [4] Asai Asaithambi, Numerical solution of the Burgers’ equation by automatic differentiation, Appl. Math. Comput. 216 (2010) 2700–2708.
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